Math 321 - Introduction to Applied Mathematics
General information about the course
can be found in the general course page and
won't all be repeated here.
Plan for course
It is intended to cover the whole book. The pace of the course should be
the 2006 version.
The course aims to introduce students to the art of building and
refining mathematical models. This is also the main theme of
the exercises in the textbook, which will be used as a framework
for class discussion. However, this noble aim is
difficult to measure, so exams will concentrate on
mathematical techniques. To practice these current techniques,
there will be a quiz in each class (unless there is an exam
scheduled for that date).
Two exams are planned, one at the end of the first two parts of the
text. The final exam will revisit parts 1 and 2 as well as including
problems based on part 3.
Fall 2008 Information
The instructor is Prof. Bumby (follow this link for
home page). Contact information and office hours can be found on
his home page.
During the semester, a log was kept on the Sakai site. Here is a copy
of that log.
- September 03: Sections 1–4. Using physical laws to write
equation for forces. Newton's laws connect forces with
acceleration, which is the second derivative of displacement, so
differential equations are inevitable. When the spring is
vertical, gravity appears in the equation giving a nonhomogeneous
equation. We appealed to section 6 to test that all terms in
an equation represented forces.
- September 08: Sections 5–9. Various background topics,
including exercise 8.2 leading up to a study of spring-mass
systems with one or two springs following the example in the
- September 10: Sections 10–13. Damped oscillations. If
there is a a force proportional to velocity and opposite to the
direction of motion, the equation remains linear but the solution
contains an exponential factor with a negative exponent. For small
values of the resisting force, there is still oscillation, but
with a longer period than in the undamped case.Exercise 12.3 was
discussed: it dealt with the periodicity of relative
maxima,.For large values of the resisting force, the solution is a
sum of two exponentials, with the slower decay dominating.
Exercise 13.4 was discussed: it dealt with the behavior as the
resisting force goes to infinity.
- September 15: Sections 14–16.Basic properties of the
pendulum. Setting up the problem is easiest in a coordinate system
that moves with the pendulum and has one axis that points in the
direction of the rod connecting the weight to the pivot. The
slightly more general approach of studying an arbitrary motion
using polar coordinates is sketched in exercise 14.2. Exercise
15.1 was discussed: it used Taylor's theorem to estimate the error
in replacing the sine of theta by theta.
- September 17: Sections 17, 18 and a look ahead to 22. There
was also a brief discussion about whether a small relative error
in the function of x giving d2x/dt2 means
that the solutions x(t) of an equation with the error will
approximate one without. In some places, like exercise 15.1, the
text suggests this. However, a more careful study of differential
equations suggests that this is only a local truncation error, and
the compounding effect of previous errors needs to be considered.
This second type of error typically increases exponentially with
time. In section 17, constant solutions of a differential
equation were considered, and the question of stability was
introduced. In section 18, a linear approximation was proposed as
a test for stability. This is best considered by introducing a
phase plane, which appears in section 22. Eliminating t from the
system leads to an equation relating displacement and velocity
that is separable for any equation for which acceleration is a
function only of position. Solving this equation gives a relation
between position and velocity that is itself a differential
equation. This equation may be interpreted as conservation of
energy as in sections 19 and 20.
- September 22: Sections 19, 20. Conservation of energy as a
unifying idea for describing the trajectories in the phase plane.
Introducing integrals to measure time along a trajectory.
- September 24: Sections 26–28. Basic properties of
damping in the nonlinear case. Energy in the system is no longer
conserved; it decreases because some escapes as heat. Energy
curves are still useful although they are no longer trajectories.
Isoclines are defined and used to get some qualitative properties
- September 29: Guest lecture by Professor Wheeden featuring
Exercise 19.7 on Escape Velocity. Otherwise an
opportunity for a fresh view of all of part 1.
- October 01: Sections 30–34. The first two of these
sections give an overview of problems of population growth. The
mathematics begins when we explore the consequences of assuming
that the change in population over a fixed time interval is
strictly proportional to the population at the start of that time
interval. This is a simple example of a difference equation.
There are many common features between difference equations and
differential equations including the ability to find solutions by
assuming an exponential solution and exploiting linearity. Other
topics mentioned were compound interest and the Fibonacci sequence.
The quiz on this material will be on October 8 because of the exam
on October 6.
- October 06: Exam on part 1.
- October 08: Sections 35–36. Two variations on the simple
population model are considered. In one, the population is
classified by age with different birth and death rates for
different ages (in the case of birth rates, the age of the parent
is considered). In the second model, probabilistic considerations
- October 13: Sections 37–38. The logistic
equation. To avoid the problem that the simple model has an
unstable balance between unsustainable exponential growth and
exponential decay leading to extinction, we seek a model showing
exponential growth when the population is small, but with a growth
rate that decreases as the population decreases. The simplest
example has a rate that is a linear function of population. This
is the logistic model. By choosing a linear function that is zero
for the expected maximum population, you get an equation that is
sure to justify the limit you expect. For a first order
autonomous equation, there is a phase line. The
equilibrium points are identified, and between them, solutions to
the equation are either steadily increasing or steadily
decreasing. To find out which, simply evaluate the right side of
the equation at an interior point of each interval between
equilibrium points. Alternatively, the equation can be
linearized at each equilibrium point. If this shows the
point to be attracting, the arrows on adjacent segments point
toward the point; if repelling, they point away. The two methods
must give consistent results. In higher dimensions, only the
linearization at equilibrium points will be available, so it is
useful to relate this method to a more elementary method in a
simpler case so you can begin to trust it.
- October 15: Sections 39–40.Just to show that it is
possible, we obtain the explicit solution of the logistic
equation. One feature that is easily seen from this is that the
decreasing solutions all have vertical asymptotes.
- October 20: Sections 41–42. In order to learn more what
to expect about the time-delay model, we linearize at the
equilibrium point. The corresponding linear difference equation
can be solved exactly, and the question of stability can be
answered using this solution.
- October 22: Sections 43–47. These sections review
properties of a system of two linear differential equations.
- October 27: Sections 48–53. The predator-prey
- October 29: Section 54. The competing species model.
- November 03: Sections 56–57. Introduction to traffic flow
and properties of a velocity field.
- November 05: Sections 58–59. Traffic flow and traffic
- November 10: Exam on Part 2.
- November 12: Sections 60–63. Two fundamental principles
of the study of traffic flow are introduced. First, the
assumption that cars are neither created nor destroyed in the
daily commute leads to a conservation law that can be expressed
as a partial differential equation (parsed as an equation
involving partial derivatives). Second, the mindless nature of
the daily commute is modeled with an assumption that the speed of
a car is determined only by the density of traffic. This leads to
a velocity field describing the speed of a car in terms of its
position in time and space. As we have seen, a velocity field is
an ordinary differential equation whose solution describes the
behavior of individual cars. The traffic flow also becomes a a
function of density that is zero at both zero density and the
maximum (bumper to bumper) density). We confine attention to the
case in which this function is concave downward, so that it has a
unique maximum, although no explanation is proposed for why the
relation between density and speed would lead to this behavior of
- November 17: Sections 65–67. Section 64 was skipped
since it is not used later, and the details of Section 66 were
postponed. Some general properties of partial differential
equations are given, and the nature of the solution of some linear
equations arising in the study of traffic flow is described.
- November 19: Sections 66–69. The fundamental equation of
traffic flow has the property that any constant function is a
solution. By considering small perturbations of such solutions,
we find that the perturbations must approximately satisfy a linear
equation. The solution described in Section 67 produces density
waves that travel forward in light traffic and backward in heavy
traffic. The wave pattern would seem fixed to an external
observer moving at the wave velocity although different cars would
be in the wave.
- November 24: Sections 70–73. The semi-infinite highway
gives an example in which there is a boundary condition in the
space variable as well as the initial condition in time. In light
traffic, the solution to the equation splits into two cases: at
any time, the more distant densities are the evolution of the
initial density, while those near the entrance are determined by
the boundary condition. The solutions of the linearized equation
were constant on certain lines, corresponding to an external
observer moving at constant velocity. These lines are called
characteristics. In general, a characteristic is a curve on which
the partial differential equation reduces to an ordinary
differential equation. The characteristics in this example are
more special: the solution is constant of the characteristic and
the characteristic is a straight line. In the example of a
traffic light turning green after a long stream of cars has built
up behind it, the characteristics that don't correspond to maximum
velocity or zero velocity all pass through the origin. For any
assumption of the dependence of velocity on density, this allows
the density at each point to be determined. Once the density is
known, methods used at the beginning of this part of the text
determine the motion of individual cars. In Section 73, this
solution is found when a linear relation between density and
velocity is assumed.
- December 01: Sections 77–78. The examples of sections
74–76 will be skipped to get directly to the idea of a
shock. In previous examples, we were careful to make sure that
characteristics did not intersect, but we can't always be so
lucky. In particular, traffic stopping for a red light gives an
important example in which the characteristics from the original
motion of traffic and those from the condition at x=0 for positive
t overlap in a large portion of the (x,t) plane. One one set of
characteristics, density is the original uniform density on the
highway; on the other set, it is the maximum density. Only one of
these can be valid at any given point. To resolve this ambiguity,
the integral form of the conservation law is considered instead of
the differential form that we have been using. This leads to
discontinuities in density being allowed only when the resulting
ratio of the change in flow to the change in density is equal to
dx/dt on the curve, called a shock, on which the value
changes. In the case of a red light, there are two constant
values to separate, so we get a line of fixed slope as the shock.
Among all such lines, the shock is the one passing through tho
origin, since this in the only place where the given boundary data
last revised by
RT Bumby on June 19, 2009